Dense Set - Density in Metric Spaces

Density in Metric Spaces

An alternative definition of dense set in the case of metric spaces is the following. When the topology of X is given by a metric, the closure of A in X is the union of A and the set of all limits of sequences of elements in A (its limit points),

Then A is dense in X if

Note that . If is a sequence of dense open sets in a complete metric space, X, then is also dense in X. This fact is one of the equivalent forms of the Baire category theorem.